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Theorem hlbn 19307
Description: Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Assertion
Ref Expression
hlbn  |-  ( W  e.  CHil  ->  W  e. Ban )

Proof of Theorem hlbn
StepHypRef Expression
1 ishl 19306 . 2  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
21simplbi 447 1  |-  ( W  e.  CHil  ->  W  e. Ban )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   CPreHilccph 19119  Bancbn 19276   CHilchl 19277
This theorem is referenced by:  hlcms  19310  hlprlem  19311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-hl 19280
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